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By James E. Humphreys (auth.)

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Z, the origin. Among 1 ~ P, those vectors y -w I if necessary Clearly s L then Wl+W2-W first triangle, which is absurd. P is contained in C = ~2. by w I, w 2 w I = ml2z L + m22 independent such contains in L closest of wI w 2 =mllZ to choose one + m21. and by replacing det y > O. ( L wI By construction, no other point of by the L; would be an interior point We conclude has no point in a fundamental domain of L if determined by of the that the p a r a l l e l o g r a m in its interior, for the But the latter domain has volume of the p a r a l l e l o g r a m y ~ F of the triangle with vertices w I, w 2, Wl+W2, determined find span the complex plane.

2 to find domain for X ~ K~ K in (v ~ s) £ 1/6 v sum to XE of again an the contains element adele a 0 (v g S), of a XE whose (v / S, v ¢ v 0) fundamental and an are to yield the desired As before, We continue K the field, to use the notation of e K. given av each a e AK is the This can be applied (v ~ S) or D THE M U L T I P L I C A T I V E is a number so GROUP of degree n = r+2s I, following Cassels over Q. [1, ~16-18]. Ideles Idele topology Let U v = {a Uv = Ov - Pv [ iv 6 5 ~ K ~7. 1 domain, element components III.

Its usual [Hint: topology. ] that lemma, c J~ is closed we recall in A K. 1, In particular, any finite LEMMA. if av ~ 0v since on us by Haar measure con- Card k v = index of but ~v ~ Uv' idele and adele then Pv in

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